Quasiperiodic Skin Criticality

• Quasiperiodic skin criticality is a phenomenon in non-Hermitian quasicrystals where multifractal eigenstates concentrate at boundaries due to nonreciprocal hopping and quasiperiodic modulation.
• Analytic methods based on nonunitary gauge transformations yield energy-independent scaling laws, with the inverse participation ratio scaling as N^(-β) and universal critical exponents governing phase transitions.
• The framework generalizes to multiband, long-range, and driven lattice models, offering robust experimental signatures in photonic, acoustic, cold atom, and electronic circuit platforms.

Quasiperiodic skin criticality is a universal phenomenon emerging in non-Hermitian quasicrystals wherein the non-Hermitian skin effect (NHSE) and multifractal criticality coexist in a robust and analytically tractable fashion. In these systems, the spectral and spatial structure of eigenstates defies the traditional dichotomy of extended and localized states, yielding phases where critical multifractality is locked to a boundary-localized skin profile. This regime arises generically in one-dimensional lattices with quasiperiodic modulation and nonreciprocal hopping, and it persists in multiband generalizations, long-range hopping models, periodically driven lattices, and systems with additional spin-orbit interactions. The interplay of non-Hermiticity, topology, and quasiperiodicity enables a precisely solvable framework for analyzing mobility edges, phase transitions, and scaling exponents, distinguishing quasiperiodic skin criticality as a new universality class in non-Hermitian condensed matter physics (Chen et al., 30 Jan 2026, Peng et al., 2024, Tong et al., 24 Jan 2025, Liu et al., 2020).

1. Foundational Models and Analytic Structure

Quasiperiodic skin criticality is most transparently realized in modulated non-Hermitian Hatano–Nelson (HN) chains and generalized non-Hermitian Aubry–André–Harper (AAH) models. The essential ingredients are:

• Quasiperiodic Modulation: Lattice hopping amplitudes and/or onsite potentials are modulated by an irrational frequency β (commonly β = (√5–1)/2).
• Nonreciprocal Hopping: Asymmetric hopping amplitudes t_R ≠ t_L induce the NHSE; nonunitary gauge transformations facilitate analytic tractability.
• Special Inverse-Hopping Construction: t_R(j)=J cos[2πβ (j+½)+φ], t_L(j)=1/t_R(j), ensuring t_R(j)t_L(j)=1 per bond, forms a non-Hermitian, locally reciprocal, and quasiperiodic model (Chen et al., 30 Jan 2026).

Via a site-dependent, diagonal nonunitary transformation, these models are mapped onto clean, Hermitian tight-binding chains, allowing for closed-form analytic solutions for the full spectrum and spatial profiles of eigenstates. The general wavefunction amplitudes are given by:

$$|\psi_j^{(m)}| \propto \exp\left\{\Gamma_j - \frac{j}{N} \Gamma_{N+1}\right\}$$

where $\Gamma_j = \sum_{\ell=1}^{j-1} \ln t_R(\ell)$ depends only on the global phase φ and system size N.

2. Skin Effect, Multifractality, and Energy-Independent Critical Envelopes

A defining characteristic of quasiperiodic skin criticality is that all eigenstates share an identical multifractal spatial envelope, regardless of their eigenenergies. This is quantified by the inverse participation ratio (IPR):

$$\textrm{IPR} = \frac{ \sum_{j=1}^N \exp\left[4 \Gamma_j -4(j/N) \Gamma_{N+1}\right] } { \left[ \sum_{j=1}^N \exp\left[2 \Gamma_j -2(j/N) \Gamma_{N+1}\right] \right]^2 }$$

The IPR demonstrates strict energy-independence and is controlled solely by a global phase φ. The IPR scales as $N^{-\beta}$ with $0<\beta<1$, revealing that eigenstates are critical in the sense of multifractality: extended states would yield β=1 and localized states β=0. The universal exponent β is set by the distribution of $\ln|\cos(2\pi\beta j+φ)|$ and is numerically β ≈ 0.61 for β=(√5−1)/2.

Skin criticality furthermore mandates that all bulk eigenstates pile up at one boundary (the NHSE) while inheriting a fractal, scale-invariant spatial envelope. This persists across multiband extensions, with proper inverse-hopping constraints on each leg yielding identical scaling and multifractal exponents.

3. Generalizations: Multiband, Long-Range, and Driven Lattices

Multiband and Ladder Realizations

Analytically exact skin-criticality extends to ladder-type (multiband) HN/AAH models (Chen et al., 30 Jan 2026). Provided each chain observes the inverse-hopping condition, the nonunitary transformation again yields a clean multiband model. In symmetric cases (e.g., identical coupling strengths across bands), the multifractal profile and IPR coincide with the single-band result, up to minor normalization differences.

Long-Range Hopping

In non-Hermitian AAH models featuring power-law hopping $t_s \sim 1/s^a$, the skin effect is continuously weakened as the hopping range grows (i.e., as $a\to 0$) (Peng et al., 2024). The skin localization length scales as $\xi(a)\sim a^{-\gamma}$ for $\gamma\approx 1.0$–1.2, implying that long-range hopping counteracts the NHSE by enabling more direct transport paths that bypass the boundary accumulation. The transition from ergodic to skin states is tightly linked to mobility edges associated with quasiperiodic-induced fractal band-structure.

Time-Dependent and Floquet Systems

Periodically driven (Floquet) non-Hermitian quasiperiodic lattices introduce new routes to skin criticality: adiabatically modulated fields can destroy conventional Wannier–Stark localization and generate regimes with multifractal skin states and multiple mobility edges (Chakrabarty et al., 2024). The skin effect is robustly restored by periodic rather than static driving, and the resulting skin states are necessarily multifractal, in contrast to ballistic, skin-type localization seen in static systems.

4. Scaling Laws, Universality, and Critical Behavior

The scaling behavior at skin-criticality is characterized by universal exponents. The IPR and higher-order multifractal dimensions $\{D_q\}$ for all eigenstates in the skin-critical phase are energy-independent and dictated by the spatial modulation:

$$\textrm{IPR} \sim N^{-\beta}, \qquad D_q = \frac{1}{1-q} \lim_{N\to\infty} \frac{ \ln \sum_j |\psi_j|^{2q} }{ \ln N }$$

Transitions into and out of the skin-critical regime universally exhibit correlation length divergence with exponent ν=1; e.g., localization length $\xi\sim|\lambda-\lambda_c|^{-1}$ and similar power-law scaling of bidirectional skin order parameters in extended models with bidirectional NHSE (Liu et al., 2020, Padhi et al., 2024). The onset of bidirectional skin effect (both edges) as disorder or potential strength is tuned exemplifies critical behavior analogous to that in second-order phase transitions.

Topological invariants, specifically spectral winding numbers calculated under periodic boundary conditions, sharply diagnose transitions between skin, skin-critical, and localized regimes. The critical points coincide with jumps in winding number, enforcing a topological underpinning for skin criticality (Jiang et al., 2019, Tong et al., 24 Jan 2025).

5. Variants, Modifications, and Non-Universal Features

Extensions beyond the baseline models reveal a rich variety of non-universal phenomenology:

• Spin-Orbit Interactions: Rashba-type couplings modify the skin effect's directionality and criticality. The critical point is renormalized to $\lambda_c=2\sqrt{t_R^2+\alpha_y^2+\alpha_z^2}$, and strong spin-flip terms can partially suppress or even reverse skin accumulation (Chakrabarty et al., 2022).
• Next-Nearest-Neighbor Hopping: Adding NNN terms to the HN or AAH model with quasiperiodicity induces bidirectional skin effect: eigenstates localize at both ends, and the direction of skin effect can be reversed by tuning the model parameters. Skin-criticality in this case manifests as the onset of bidirectional NHSE and is associated with its own set of critical exponents (Padhi et al., 2024).

Variants with spatially modulated, nonreciprocal hoppings transform both extended and critical phases into skin phases under OBC, generating skin-critical envelopes with fractal structure not present in Hermitian or reciprocity-preserving systems (Tong et al., 24 Jan 2025).

6. Experimental Signatures and Realizations

Physical realization of quasiperiodic skin criticality is accessible through several platforms:

• Photonic Lattices: Engineered flux-modulated couplings permit direct observation of skin-accumulated, critical intensity envelopes (Chen et al., 30 Jan 2026).
• Acoustic Metamaterials: Tunable nonreciprocity and quasiperiodicity in acoustic meta-chains allow in-situ variation of the global phase φ and real-space imaging of multifractal skin modes.
• Cold Atom Experiments: Dynamically generated optical lattices implement driven, non-Hermitian, and quasiperiodic baths. Monitoring local density profiles enables the study of universal skin-critical behavior.
• Electronic Circuits: Sci-fi skin effect and skin-critical transitions have been mapped onto RLC networks with parameter-matched inductors, capacitors, and resistors, providing a testbed for the boundary amplification and critical scaling predicted by analytic theory (Jiang et al., 2019).

Robust, energy-independent multifractal envelopes and boundary-selective amplification are direct observables, as is the divergence of localization (skin) lengths near transition points.

7. Outlook and Theoretical Implications

Quasiperiodic skin criticality demarcates a new universality class for non-Hermitian quasicrystals, typified by analytically solvable models with energy-independent, multifractal skin profiles. The phenomenon exposes deep connections between bulk-boundary correspondence, topological invariants, and the interplay of disorder, quasiperiodicity, and nonreciprocity. The analytic tractability—enabled by nonunitary gauge maps—permits rigorous derivation of phase diagrams, mobility edges, Lyapunov exponents, and critical exponents across a wide arsenal of models.

Ongoing research aims to generalize these insights to higher dimensions, interacting systems, and nonequilibrium steady states, investigate the interplay with other symmetry classes, and clarify the dynamical signatures in quantum and classical setups with tunable non-Hermiticity (Chen et al., 30 Jan 2026, Peng et al., 2024, Chakrabarty et al., 2024, Liu et al., 2020, Padhi et al., 2024, Tong et al., 24 Jan 2025, Jiang et al., 2019, Chakrabarty et al., 2022).